Perturbations of quadratic hamiltonian systems with symmetry

A NNALES DE L ’I. H. P., SECTION C

E MIL I VANOV H OROZOV

I ^LIYA D ^IMOV I ^LIEV

Perturbations of quadratic hamiltonian systems with symmetry

Annales de l’I. H. P., section C, tome 13, n

^o

1 (1996), p. 17-56

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Perturbations of quadratic

Hamiltonian systems ^with symmetry 1

Emil Ivanov HOROZOV

Iliya

Dimov ILIEV

University ^of Sofia, Faculty of Mathematics and Informatics, 5, J. Bourchier blvd., 1126 Sofia, Bulgaria.

Institute of Mathematics, Bulgarian Academy ^of Sciences, P.O. Box 373, ¹⁰⁹⁰ Sofia, Bulgaria.

Vol. 13, ^n° 1, 1996, p. 17-56 Analyse ^{non linéaire}

ABSTRACT. - It is

proved

^that

arbitrary quadratic perturbations

^of

quadratic

Hamiltonian

systems

^{in the}

plane possessing

^central

symmetry

can

produce

^{at most two limit}

cycles.

Key ^words: Bifurcation, ^limit cycle, quadratic ^vector field, ^Abelian integrals.

RESUME. - Nous prouvons que les

perturbations quadratiques

arbitraires des

systemes

hamiltoniens

quadratiques

^et

symetrie

^{sur la}

plane peut produire

^au

plus

^deux

cycles

^limites.

0. INTRODUCTION

In this paper ^we prove the

following

^result:

1 Research partially supported by Grants MM - 402/94 and MM - 410/94 of the NSF of

Bulgaria.

Annales de l’lnstitut Henri Poincaré - Analyse ^non linéaire - 0294-1449 Vol. 13/96/01/$ 4.00/@ Gauthier-Villars

THEOREM 1. - Let H be a

generic

^cubic Hamiltonian with a central symmetry :

H(-x, -~) _ -H(x, y).

^{Then any}

quadratic perturbation of

the

corresponding

Hamiltonian system

has at most two limit

cycles for E

^small

enough.

Our interest in

systems (0.1)

^{was motivated}

by

^the

project

^{to obtain an}

explicit

^and

sharp

^bound

c(2)

for the number of limit

cycles

ⁱⁿ

quadratic

vector fields which are close to Hamiltonian ones or at least to obtain an

explicit sharp

bound for the so called Hilbert-Arnold

problem [2]

^{for n = 2}

(see [9], [12], [15], [16], [27]

for results in this

direction).

The Hilbert- Arnold

problem

^is formulated as follows: find an upper bound

Z(n)

^for

the number of isolated zeros of the

integral

where

b ( h)

is the oval

(compact

^connected

component)

^{of the}

algebraic

curve

H(x, y)

^{= hand}

deg H

^{= n +}

1, deg f ,

^{g = n}

^(or

^more

generally,

find ^an upper bound

Z(n, m) provided max (deg f , deg g)

⁼

^m).

Integrals

^{of this}

type

in studies of limit

cycles

^were introduced

by Pontrjagin [23].

The first nontrivial case was considered

by Bogdanov [6].

The existence of an upper bound

Z(n)

^was established

by

A. Varchenko

[25]

and

Khovanskii _[20].

But

making

this bound

explicit

^{seems to be a difficult}

problem. Quite recently,

^Yu.

Il’ yashenko

and S. Yakovenko

[19] proved

^a

double

exponential

estimate for

Z(n, m) provided H

^{is a} fixed Hamiltonian from some dense set of

generic

Hamiltonians:

Z(n, m) 22~~m~ . _Although

this bound is far from the

expected (polynomial)

^one, up ^{to now} this is the best known result.

More

generally,

^{one can consider}

perturbations

^of

integrable systems

instead of Hamiltonian ones. In the

quadratic

^{case, an essential}

part

^{of the}

problem

also consists

(see [28])

ⁱⁿ

counting

the isolated zeros of certain

(not necessarily Abelian) integrals. Zol~dek

^has

proved [28]

^{that in}

quadratic perturbations

of Lotka-Volterra

systems

^{with a center the}

corresponding integrals

^{will have} at most two zeros.

Among

^the

remaining

three classes of

quadratic systems

^{with a} center, ^most of results have been obtained for the Hamiltonian case

(see

e.g.

[17]).

^Almost

nothing

is known about the other two

classes,

^{where at} least three limit

cycles

^can appear.

Annales de l’Institut Henri Poincaré - Analyse ^{non linéaire}

In a recent paper

[ 16]

^{we found that for}

generic

^cubic Hamiltonians with three saddles and one centre the exact value for both

c(2)

^and

Z(2)

^{is two.}

For the two

remaining

basic classes of cubic Hamiltonians

(namely

^those

with one saddle and one centre and

respectively

^{with both two} saddles and

centres)

^the

problem

is still open. Our

conjecture

is that the exact bound for all

generic

^{cases is two}

(the

^{same bound was}

conjectured

ⁱⁿ

[28]).

^The

symmetric

Hamiltonians form a co-dimension one subset within the class of Hamiltonians

having

^two saddles and two centres. For many ^reasons it seems to us that the

symmetric

^{case is the one}

allowing

^{to be treated}

comparatively

^{easier at least as far as the}

computations

^{are concerned.}

In the statement of Theorem 1 as well ^as in the above comments we use the notion of

"generic"

Hamiltonian. There are several definitions of this notion even for

quadratic

Hamiltonian vector fields.

Throughout

^this

paper ^we say the cubic H is

generic

^{if the}

corresponding

Hamiltonian

vector field dH ⁼ 0 has a centre and does not

belong simultaneously

to any of the other

integrable

classes of

quadratic

^{vector fields}

(as

^listed

in

[28]).

The latter condition has a

simple geometrical interpretation:

^non-

generic

^are

exactly

^those Hamiltonians for which the level curves in suitable coordinates have an axis of

symmetry.

On its hand

evidently

the central

symmetry

^{of H}

geometrically

^{causes dH = 0 to be}

centrosymmetric

with

respect

^{to a suitable}

point (xo,

^{which we} without loss of

generality

have chosen to coincide with the

origin.

^The

general

^case

H(xo - x, y) = -.H(~ -

is dealt with a mere translation.

One can

easily

check that after eventual linear

change

^{of the variables}

each cubic Hamiltonian with a central

symmetry

^and

having

^{a centre can}

be written in the form

Then

"generic"

^{in this case}

means ~

^{0. Selected level curves of H for}

/1 ⁼

0, _~c

^E

(0,1)

^{and /1 = 1 are drawn in}

Fig.

^1.

System (0.1 )

has attracted the interest of several other authors

[4], [9], [21].

^In

particular

^Bamon

[4]

^{found a value of} /1

(~c

⁼

(4 3 -1) / (4 3 + 1)

⁼

0,227....) and

^{a suitable}

perturbation

for which around one of the foci a

saddle

loop

^{and at least one limit}

cycle

^can coexist. Drachman et al.

[9]

proved

^that

for

^{close to} 1 and for a

specific perturbation ( f , g)

^the

system (0.1)

^{has two limit}

cycles

^{around one} of the foci and has no limit

cycle

around the other one.

(In [4], [9]

^a different coordinate

system

^was

used).

Our paper ^covers all

generic symmetric

Hamiltonians and all small

quadratic perturbations showing

^{that the}

sharp

estimate of the total number of limit

cycles

ⁱⁿ

(0.1)

^{is two.} Moreover from our

analysis

it follows that

Vol.

all numbers 2 and

positions

of the limit

cycles permitted by

^the

general theory

^of

quadratic systems [7]

^{can be realized. It is} also easy ^to describe all bifurcations when the

parameters

^of

( f , g)

vary. Some of these results

(for special perturbations)

^were announced in

[21] ] but apparently

^without

proofs.

In this paper ^we do not consider the

degenerate

case ⁼ 0 which

requires

^an

analysis

up ^{to a} fourth order in c. This case as well as the other

(nonsymmetric) degenerate

cases are an

object

of another research

(in progress).

Perhaps

it is worth

noticing

^that

system (0.1)

falls into class

IIIa~o

^{of the}

Chinese classification

[26].

^More

precisely

this class consists of the

systems

When

b = l = m = b = 0, n ~

^{0 the}

^{system (0.4)}

is Hamiltonian and

centrosymmetric

^with

respect

^{to the}

point ( - 2a , 2n ) .

^{Then a}

^simple

corollary

from Theorem 1 is the

following

^one:

COROLLARY 1. -

In I ^~

^{0 and small}

^{b, l,}

^m;

^b,

^system

(0.4)

has ^{no more} than two limit

cycles.

The

techniques

^{we use to}

get

the results of the

present

paper is ^a combination of the

techniques

^from

^[16]

^{with some ideas from}

^[11].

^In

particular

^we

extensively

^use the notion and the

properties

^{of the centroid}

curve.

(A

^{centroid curve is formed}

by

^{the mass centres of a continuous}

family

^{of ovals within the level curves of}

^H.)

^{There are two centroid}

curves in the considered case,

corresponding

^{to two}

periodic

annuli. In our

study

^the

elementary

^fact

applies

that each intersection

point

^{of the zero}

divergence

^{line in}

^(0.1)

^{with a centroid curve}

corresponds

^{to a zero of}

I ( h)

and hence to a limit

cycle. Using

the mutual

position

^{of the two centroid}

curves, Theorem 1 is

easily

^{seen to be a} consequence of the fact that in the

generic

^case each of the centroid curves has a non-zero curvature.

A

convexity argument

^is

quite

natural in situations where the

expected

number of

cycles

^{is two.}

Incidentally,

^a

convexity

idea has been used

by Il’ yashenko ^[18]

ⁱⁿ his paper about Abelian

integrals arising

ⁱⁿ

symmetric perturbations

of Hamiltonian

systems

^with

symmetry

^{of order two.}

A more detailed sketch of ^our construction which ^we believe will

help

. the reader to overcome easier the rather technical

proof

^is

given

^{in the}

next section.

Annales de l’lnstitut Henri Poincaré - Analyse ^{non linéaire}

1. PRELIMINARIES AND OUTLINE

OF

THE PROOF

We start with the definition of the centroid curve for _{any cubic} Hamiltonian H

having

^{a centre. It is} easy ^{to see} that the

integral I(h,)

from

(0.2)

^{can be written} in the form

where Int

(h)

^{is the}

region

^{inside the}

oval 6(h)

^{and = -}

fx-gy.

Define the

integrals

Then the centroid

point [10]

^of

region

^Int

(h)

^has coordinates

(~(h,), ~(ja))

where ~

^{= =}

Y/M.

DEFINITION 1.1. - Let

(hl, h2~

be the maximal interval of existence of the oval

~i(h) (including

^the

separatrix cycle

^{and the centre}

inside).

^Then

the curve

is called

[16]

the centroid curve of the

family

^{of ovals.}

The

importance

^{of the}

_concept

^{of the centroid curve lies} in the fact that its

geometry

contains the

complete

information about the

possible

^{zeros of}

each

integral

⁼

aX(h)

^{+ +}

^(i.e.

^{for all values} c~,

,~,

~y and all

perturbations f, g), although

^the definition of L

depends only

^on

H. We mention also the

following

obvious facts:

a)

^{L is affine}

invariant; b)

the

endpoints

^{of L are} the centroid Z of the area bounded

by

^the

separatrix cycle

^{and the centre C}

lying inside; c)

^for

non-generic Hamiltonians,

^L

is a line

segment; d) M ( h ) gives

^{the area of and T = M’ is}

the

period

^{function. It has been}

recently proved [8]

^{that T is monotone}

and hence

M~~ ~

^0.

Specifying

Definition 1.1 to the case of

symmetric

Hamiltonians

(0.3)

^we

see that there are two continuous families of ovals

8 (h)

^defined

respectively

in the intervals

~-h~,

^{and where 0}

hs h~

^are

given by

the formulas

Vol.

We shall denote

by ^Li

^and

by L2

^the

corresponding

^{centroid curves. We}

point

^{out that in view of the}

symmetry

^{of H the curve}

L2

^is

just Li

^rotated

on an

angle

^{In the}

sequel

^{we use L to} denote any of

Ll, L2.

^Further

denote

by Ci, C2

^{the centres and}

by Sl , 82

the saddles of the Hamilto- nian

(0.3).

^{For later use we}

_compute

their

coordinates, obtaining

and

respectively

⁼

(xs, ~S) _ (-~c, -~~), C2

⁼

-G’~, ~‘2

⁼

The

following

^{theorem sums} up several

properties

of the centroid curve

of

generic

Hamiltonians obtained in

[16]

^{and which we} need in the

present

paper.

THEOREM 1.1. -

Suppose

^{that H is}

generic.

^Then

(i)

^{Near its}

endpoints

^{the curve L is}

regular,

^i.e.

(h)

⁺

r~’2 (h) ~

^0.

(ii)

^{Near the}

endpoints of

^{L the curvature » is not zero} and has the same

sign.

It is not hard to see

[ 16]

^{that for a}

given perturbation f, g

ⁱⁿ

(0.1)

^with

( -I- ~,~ j ~ ^0,

the number of limit

cycles

ⁱⁿ

^(0.1)

for small c is

equal

^{to the}

number of the intersection

points (counting

^the

multiplicities)

^{of L with the}

line £ : ax +

-~- ~y

⁼

0, provided

^{L is}

regular.

^{Moreover as c --~ 0 these}

limit

cycles

^{tend to the ovals}

8(h)

^{whose centroid}

points

^{lie on £. Hence}

our

goal

^{would be achieved if we show that}

Li, L2

^are

regular

and that any line intersects their union L ⁼

L1

^U

L2

ⁱⁿ at most two

points.

In section 5

we prove the

regularity

^{of the centroid curves in the case of an}

arbitrary generic

Hamiltonian with two centres,

using

Picard-Lefschetz

theory [3].

The

proof

^{is similar to that in}

[16]

^and

exploits

ideas from

[12]

^and

[22].

Note that the

regularity

^{of L as well as the}

nonvanishing

^{of the curvature}

are affine invariant conditions. Most of our efforts will be concentrated to show that an

arbitrary

line £ intersects

Li

^{in no more than two}

points.

Therefore the

following

^theorem is the bulk of our construction:

THEOREM 1.2. - The curvature

of

the centroid ^curve

Ll

^{at each}

point

is non-zero.

Our

plan

^{to show the}

nonvanishing

^{of the curvature is to start with a}

Hamiltonian H for which we

already

know that. Such one can be obtained for

example

from the standard

elliptic

Hamiltonian

Ho == y2 -~- x3 - ~

via small

perturbation

in the direction of the

symmetric

Hamiltonians. This is done in section 6. From Theorem 1.1 we know that in the

generic

^case

the curvature near the

endpoints

of the centroid curve is

always

^non-zero.

Annales de l ’lnstitut Henri Poincaré - Analyse ^non linéaire

This

implies

^that

varying

^the

parameters

^{in H if the curvature}

acquires

^a

zero this would be at least a double zero. It will

correspond

^{to a zero of}

multiplicity

^at least four of

I ( h) provided f, g

^are

properly

^chosen

([16], Corollary 2.1).

^A

simple

^but

important

^fact is that the second derivative of I and the first derivative of the

period

^{function T}

satisfy

Picard-Fuchs

system

^{of order} two, ^and hence the ratio w ⁼

I" /T’

^{satisfies a Riccati}

equation.

^This

observation,

used for the first time in

[ 11 ],

^{is based on the} fact that the residua of the form w

= - ,~ dy + 9 dx

^{are linear in hand}

therefore the second covariant derivative of the Gauss-Manin connection of w has no residua. In section 2 of the paper ^{we use} these ideas to derive Riccati

equation

^{for w.} Most of the

present

paper is devoted ^{to a}

precise qualitative study

of the Riccati

equation

^satisfied

by

^{w. In}

particular

^we

explore

^the

properties

^{of the zero isocline of the}

equation.

^{On its hand to}

study

^{the zero isocline we include it in a}

family

^{of curves}

forming

^{the level}

curves of a suitable Hamiltonian. We

repeat

^{the same}

procedure

^{this time}

considering

^the horizontal and also the vertical isocline of the Hamiltonian

system

^to obtain its

phase portrait.

^{From the}

properties

^{of the zero isocline}

of the Riccati

equation

^{we find the}

possible

^{behaviour of w. The}

upshot

is that w =

I" /T’ ^(and

^hence

I ~~ )

^{has no double zeros. All this}

analysis

is contained in section 4.

We have to

emphasize

that in section 4 we do not

study w

^{for all values}

of the

parameters

a,

j3,

~y. We are interested

only

in values _a,

corresponding

^{to lines f}

tangent

^to

Li.

^{In this case w is}

subject

^to

quite

restrictive

inequalities,

^{which we} obtain in section 3. The

proof

of Theorem 1 is

completed

in section 7

by examining

the mutual

position

of the two centroid curves.

In

theory

^{the same}

plan

^{could be}

applied

^to all another cases

(including

the one studied in

[16]). Unfortunately

^the

computations

ⁱⁿ

general

^become

much more

complicated.

^In

spite

^{of that we believe that the}

plan

^{can be}

performed, eventually

^{with some} modifications. In this

respect

^the

present

paper,

apart

^{of its own}

importance,

can serve as a model for the

general

^case.

2. PICARD-FUCHS SYSTEM AND RICCATI

EQUATION

In this section we derive several differential

equations

^satisfied

by

^Abelian

integrals.

^{Our final}

goal

will be the Riccati

equation

^satisfied

by I"

For this we need to derive Picard-Fuchs

system

^{for the}

integrals Io, I_ 1, 1-2 ,

defined below.

Given h E

(-he, -hs),

^{we consider} the level curve H ⁼

xy2

⁺

3 x3 - x

⁼

^h,

^whose

^equation

^can be written also in the form

Put z ⁼ xy +

M/2

^{and define the}

integrals

We have

and

similarly Y(h)

⁼

M(h) _ - I-1.

LEMMA 2.1. - The

integrals Io, I-1, ~-2 satisfy

^the

following

system

of equations

Proof. - Express

the left and the

right

^{hand-side of the}

identity

in terms of the functions

Ik, k

⁼

0, ~ 1, ...

^This

gives

^{a connection}

between these functions:

Similarly using

Annales de l’lnstitut Henri Poincaré - Analyse ^{non linéaire}

we obtain

Using (2.3)

^{with k = -1 and}

(2.4)

^{with k = 0 we}

get

^{the third}

equation

of

(2.1).

^{In the} same manner

applying (2.2)

^and

(2.3)

^{with k = 0 and}

(2.4)

^{with k == -1 we} obtain the second

equation

^of

(2.1).

^{At the}

end,

^if

we eliminate

Ii

^and

1-3

^from

(2.2), (2.3), (2.4)

^{with k}

= -2,

^{we obtain} the first

equation.

COROLLARY 2.2. - The

integrals X, Y,

^M

satisfy

^the system

Now we are in a

position

^{to derive the Riccati}

equation

^satisfied

by w(h)

^{= Put v = w -} ~. Then we can consider that

v ⁼

I~~/M~~

^{where in}

I(h)

^we

have ~

^{= 0.} Differentiate once the first

two

equations

ⁱⁿ

(2.5)

and twice the third one to

get

From these

equations

and from I ⁼ aX eliminate first Y and then X.

After

solving

^with

respect

^to

7

and

M

we obtain

Vol.

Here for shortness we have denoted

A i /

where

From

(2.6)

^we derive the Riccati

equation

Here B ⁼

B1

+

B2. Returning

^{to the variable w = v}

-~ -y

^and

writing (2.8)

as a

system

^we

finally get

LEMMA 2.3. - The

following

Picard-Fuchs system ^is

satisfied:

this is our basic

system

^{which we will}

study

^{in the next two sections. In}

particular

^{of a}

great importance

^{for our}

analysis

will be the

properties

^{of the}

zero isocline

Fo of (2.9), given by

^the

points (h, w )

^{on the}

algebraic

^curve

(2.10)

We will often consider

Fo

^{as the}

graph

of the two-valued function

wo (h)

determined

by P ( h, wo ( h) )

^{- 0}

("+"

^is

always assigned

^to the upper

branch).

For later use denote

by Do

the discriminant of the

quadratic equation (2.10):

Annales de l’lnstitut Henri Poincaré - Analyse ^{non linéaire}

3. BASIC

INEQUALITIES

In this section we derive several

inequalities concerning

^the

parameters

in

(2.9)

and the values of

wo

^at the critical values of

H, provided

^{the line}

P : ax +

{3y + "y

⁼ 0 and the centroid curve

Li satisfy

^{some conditions}

listed below. These

inequalities

^are crucial in our

analysis

^of

(2.9).

Through

this and the next section we suppose the

following:

(U) (i) ~

^{is a}

tangent

^to the centroid curve

Li

^{at an internal}

point

~~~jL~, rl(h,~~~

(ii)

^{the curvature of}

Li

does not

change

^the

sign (but

may

vanish);

(iii) running L1,

^{£ rotates on an}

angle

less than ~r.

We start with certain restrictions for the coefficients of the line £

imposed by (U).

LEMMA 3.1. - The

coefficients

a,

,~, ~y (multiplied if needed by -1) satisfy

the

inequalities

Proof. -

^{We are}

going

^to prove that the

angular

coefficient of £ satisfies the

inequalities

If this was done and

assuming

^that

j3

^{> 0 we}

immediately get

^{the last}

two

inequalities

ⁱⁿ

(3.1).

^{Moreover the}

loop through Si

^is

placed

^{in the}

half-plane x

^> 0 and it is

easily

verified that if

(~,

^and

(.x, y2)

^are

points

on

8 (h),

^{h E}

(-h~. -ja5),

^{then ~/i + y2} 0. Therefore

Li

^{is situated in the} fourth

quadrant

and since 03B1 0

j3

^the

equality

⁺

+ -y

^{= 0} is

possible only if 03B3

^{> 0.}

In order to prove

(3.2)

^we need another normal form

of the Hamiltonian. In these coordinates

Li corresponds

^{to values h E}

[0, ~].

Having

^{in mind}

(U),

^the

proof

^{consists of}

finding

^the

equations

^{of the}

tangents

^{at the}

endpoints

^of

Li. According

^to

Corollary

^{3.4 from}

[16]

the

equation

^{of the}

tangent .~~

^of

Li

^{at the centre}

Cl(O,O)

^-Ax

(see Fig. 2).

^The

tangent .~s

^{at the other}

endpoint

goes

through

^{the saddle}

S 1

⁼

(1, 0)

^and

through

the centroid of the

loop

^area

Zi [16],

^{but the}

coordinates of

Zi

^are

given by extremely long

formulas. Because of this

we use another line instead of

.~s .

^The

saddle-loop

is determined

by

^the

equation

^{H =}

~.

The line x

= 2

divides the

region

inside the saddle-

loop

^{into two}

parts Ac

^and

s containing respectively

^{the centre and the}

saddle.

Obviously

the reflection of

As through

^{the line x}

= 2

is contained in

Ac.

^{This means}

that ~ ( 6 ) ~.

^Then

^{using (U)}

and results from

[16]

yields

^that

Zi

⁼

(~ ( 6 ) , ~ ( 6 ) )

is located inside the

triangle

with vertices

( 0, 0 ) , ( 2 , 0 ) , ( 2 , - 2 ) .

^{Let m} be the line

symmetric

^to

.~~

^with

respect

^to the line x

= 2 ,

^{m has an}

^equation

^y

= ~ ( x - 1).

^{Then the}

angular

coefficient k of an

arbitrary tangent £

^{at an interior}

point

^of

^Li

^satisfies

Annales de l ’lnstitut Henri Poincaré - Analyse ^non linéaire

Fig. ^{2. _}

the

inequalities -A ^1~

^{A. In order} to return to the coordinate

system

ⁱⁿ which the normal form is

(0.3)

^we

change

^{the variables}

(x, y)

^-

(xl,

where

In this coordinate

system

^the

equations

^of

.~~

^{and m become}

respectively (we

^{omit the}

subscript 1)

Hence the

angular

coefficient k _{of any}

tangent

^{line satisfies}

~ ~

1/~c.

Vol. 13, n°

As a direct consequence of

(2.7)

^and

(3.1 )

^{we obtain}

COROLLARY 3.2. - The constants

A; B, C, D,

^{E in}

(2.9)

^are

positive.

LEMMA 3.3. -

(i)

The values

ofw/ ( h)

^{at the ends}

of the

^interval

have the

following signs:

(ii)

^The derivatives

satisfy

Proof. - (i) Using

^the

explicit

formulas for

A, A,

^{B etc. we} express the discriminant

Do

^from

(2.11)

in the form

Also ^we have

This

gives

after obvious calculations

Annales de l ’lnstitut Henri Poincaré - Analyse ^{non linéaire}

Further,

^{with the}

help

^of

(2.7)

and the formulas for

~~

from section 1

we obtain via

straightforward computations

^that

wo ( - ~~ ~

⁼

~x~ -f-,~3~~ -~-~y.

In the same way ^we

compute

^{At the end}

where

From

(3.4)

it is clear that the

point ~~)

^is

lying

^{on the}

tangent .~~.

We will show further that

~~

xs.

Really

^{xs =} -~~ and

~~

+

5~ ^{1 - I~2.}

^{But x~ + ~/c}

5h~

^is

equivalent

^to

The last

inequality easily

follows from

h~ 9 (2

⁺

^3~) ~~ ( ~

⁺

^{~c) .}

The

inequality ~~

shows that

~e)

is. above any

tangent

^{.~ to}

Li

which

gives

^{that = +}

+ ~

^> 0. The other two

inequalities

ⁱⁿ

(i)

follow from the fact that both the centre

~~)

^{and the}

saddle

(xs,

^{are below any}

tangent

^.~.

(ii) Evaluating

^the derivatives of

wo

^{at A = 0 we}

get

Write

Do

in the form

This

gives

Calculation the values of A’

at - hs and -he yields:

Finally

^we substitute h

= - h~

ⁱⁿ the above formulas and via direct

computations

^find

Repeating

^{the above}

computation

^{for h}

= - hs

^{we obtain}

thus

proving

the lemma.

4. THE ZERO ISOCLINE OF THE RICCATI

EQUATION

In this section we list a number of

properties

^{of the zero isocline}

Fo

^of

(2.9) supposing (U)

^{hold. In}

particular

^{decisive for our} purposes will be the number and the mutual

positions

of the minima and maxima of

wo ( h) .

We start with the

following

^lemma.

LEMMA 4.1. -

(i)

^{The curve}

Fo is centrosymmetric

^with _respect ^{to the}

point (0,~).

^..

(ii) Fo

^{has no} compact components.

Proof -

^Assertion

(i)

is obvious. To prove

(ii)

^we observe that the discriminant

Do given by (2.11)

^{can have no more than two zeros for} h 0.

Together

with the behaviour near h ⁼ 0

(see

^{the next}

lemma),

^this

eliminates the existence of

compact components.

Annales de l’lnstitut Henri Poincaré - Analyse ^{non linéaire}

Next we

study

^certain

asymptotic properties

^{of h 0.}

LEMMA 4.2. -

(i)

^The

function wo (h)

^{has the}

following asymptotic expansions

(ii)

^The

function wo (h)

^{has the}

following asymptotic expansions :

Proof. -

^Direct

computations.

Our further

plan

^{is to consider the entire}

family

^{of level curves}

which is tantamount to the

phase portrait

^{of the} Hamiltonian

system

This is because the discriminant

Do

^{is either}

positive

^{or can}

change

^the

sign depending

^{on the}

parameter

^{values in}

(2.11 ). Consequently ^Fo

^{can bifurcate}

and all

possible

situations should be occurred in the

family (Fp :

^{p E}

I~~.

Again

^for

simplicity

of the notation we can consider the

portrait for 7

⁼

^0,

the

general

^case

being

^{a mere} translation w -~ _{w + ~y.}

LEMMA 4.3. -

System (4.1)

^has

exactly

^one

singular point So

^{in the}

half-plane h

^{0. The}

point So

^{is a saddle.}

Proof -

^{From the}

equations Pw

⁼

Ph

⁼ 0 eliminate w and

put h2

^{= z}

to obtain the

equation

Obviously

^this

equation

^{has at least one}

positive

^{root zo. The}

analysis

^of

the coefficient alternations and Descartes’ criterion rules out the existence

Vol.

of other

positive

^{roots. As the} Poincare index of the field in

(4.1)

^cannot

exceed

0, using

^the

symmetry yields

^that

So

^{is a saddle}

point.

Denote

by .ho == -~o

^{the abscissa} of the saddle

So .

Our further

analysis

will concentrate

mainly

^on

study

^{of two} of the isoclines of

(4.1), namely

the horizontal

© ~

and the vertical one V

= ~ Pw

⁼

0 ~ .

The

properties

^{we need are} easy ^to obtain and are summed up in the

following

^lemma:

LEMMA 4.4. -

(i)

^{The curve V is concave in the}

half-plane

^{h 0.}

(iii)

^The

asymptotics

^hold

Proof -

^Direct

computation.

’The horizontal isocline ?-~ is more

complicated

^for

studying.

Moreover it bifurcates as

parameters

vary. To describe its

properties

^{we denote}

by

~~~~~h),

^{two roots of the}

^{equation Ph (h,} w)

⁼

^0,

^{h 0.}

LEMMA 4.5. -

(i)

^Let

^{~~ -}

^{3AE > 0. Then consists}

of

^two separate

curves

given by

^the

graphs of wH.(h),

^{h 0. Their}

asymptotics

are the

following

Each

of

^the

functions w~. ( h)

^has

^exactly

^one maximum and no minima.

Let

.~2 -

3AE 0. Then the two

graphs of w~

^{meet at a}

finite point

thus

fornfng geometrically

^{one smooth curve H. For the two}

branches have the ^same

asymptotics

^{as in}

(4.2).

^The

function

^has

always

^one maximum. The

function w~ (h~

has either one minimum and one

maximum ^{or no} extrema, the latter case

including

^{also the}

possibility of

^a

collision

of

the maximum and minimum.

Proof. -

^The

asymptotics

^are

computed straightforward

^from the formulas

Annales de l’Institut Henri Poincaré - Analyse ^{non linéaire}

When B2 - 3AE ~

0 both exist for all h 0. When

B2 - 3AE

0 the discriminant D has ^a

simple

^{zero at a}

negative

^{value h =}

^h,

^i.e.

=

wH(h)

^{and both are defined for h E}

(-oo, h].

^{At the end}

computing

^the derivative of we obtain:

Putting ^h2

^{= z this}

yields

^{a cubic}

equation

^{for all extrema:}

In the case

B2 -

3AE > 0 the

asymptotics (4.2)

say that each of has at least one maximum. As

(4.3)

^has at most two

positive ^solutions,

this proves

(i).

In the case

B2 -

3AE 0 the

equation (4.3)

has either one or three

positive

solutions.

Using

^the

asymptotics (4.2)

and the behaviour near h show that has at least one maximum while either has ^no

extrema or has ^one maximum and one minimum. To obtain more

precise information,

^we observe that the derivative of

w~

^is

^negative

^{for z}

jB/2

and the derivative of

~7~

^is

positive

^{for z >}

B / 2.

^If

F ( z )

denotes the

polynomial

ⁱⁿ

(4.3),

^{we have}

F ( B ) _ - 3AE 2 ^0, F’ ( B ) _ - 8B E

^0.

Hence

(4.3)

^has

exactly

^{one zero}

greater

^than

B/2.

^This

yields

^that

wjj

^has

always exactly

^{one extremum}

(maximum).

^{Lemma 4.5 is}

proved.

We will denote the two branches of H

by ^H+

^{and ?-~- .}

Our next

objective

^{is to} locate the saddle

So

^with

respect

^{to the extrema} of Before that we need more refine

study

^of

^H,

^{which we do}

again by considering

^{the whole}

family

^{of curves}

Ph

^{= const} and their horizontal and vertical isoclines

HH = ~ Ph h

⁼

4 ~

^and

^Hv

⁼

0 ~ .

Denote also

by

?-~C~~..r

^{the two branches of}

^HH (see Fig. 3).

^{As above we} consider these curves as

graphs

^{of the}

corresponding

^functions

wH(h)

^and

wHv(h), h

^0.

LEMMA 4.6. -

(i)

^{The curves and}

Hv

^{are concave. Each}

of H±H

^has

a maximum at ⁼

6

^{and has a maximum at}

^hHV = B .

(ii)

^{The curves}

HH

^and

Hv

^{have no common}

point

^and

Hv

^{is situated}

between the two branches

of HH.

Proof -

^Direct

computation.

LEMMA 4.7. -

(i)

^{The curves V and}

Hv

^{intersect at the}

unique point of

maximum

of

^{V .}

Fig. ^3.

(ii)

^{The curves V and have no common}

point,

^{V and}

H-H

^{intersect at}

a

unique point S*

^{with an abscissa}

h* satisfying

(iii)

The saddle

So

^{is on} the lower branch

of

^H. Its abscissa

ho satisfies

Proof - (i)

is obvious.

(ii)

To prove the existence and

uniqueness

^of

^S*

we argue ^as above in

establishing

the existence of the saddle

So.

^{For the}

proof

^{of the}

inequality ^{hHH h*}

^we first take the

asymptotic

^{values of}

the functions

wHH(h)

^and

wir (h)

when ~ --~

-0, namely

Annales de l’Institut Henri Poincaré - Analyse ^non linéaire

This shows that

wHH(h)

^>

wv(h)

^{near h = -0. On the} other hand we

can see that which in details reads

or

(3B2

⁺

36C)2 4B2(4B2

⁺

6ABD). Using

^the

expressions

^for

A, B, C, D

^we

easily verify

^{that 12C}

B2,

^{B AD. Then}

(3Bz

⁺

36C)2 ³⁶⁸⁴ 4B2(4B2

⁺

SABD) 4B2(4B2

⁺

6ABD).

This proves

(4.4).

(iii)

^{We follow}

essentially

^{the same}

argument, showing

^that

This condition is

equivalent

^to

(B2 -~-12C)2 8B2(8B2 +

^{9ABD -}

6AE)

which follows from 12C

B2

and the

easily

^verified

inequality

^AE

^B2.

The

inequality ^(4.6)

^we

proved implies

^{that V} and the lower branch H- intersect at a

point So

^{with an abscissa}

ho satisfying ^(4.5). Really

^if

B2 -

3AE > 0 then

(4.2)

and Lemma 4.4

(iii) give

^{that >}

wv (h)

near h ⁼ -0. If

B2 -

3AE 0 then

(4.6) implies

^that

hHv h,

^{where h}

is the zero of D from Lemma 4.5

(ii).

^{Hence =}

wHv(h)

^>

wv(h)

because h >

hHv

^>

hv

^and

wHv(h)

^>

wv(h)

^{for h >}

hv,

the latter a

consequence of

(i),

^{Lemma 4.4}

(ii)

^{and Lemma 4.6}

(i).

^Thus

hHv ho ^h

and

(iii)

^is

proved.

Another

property

^{of H and V we} need is the

following

LEMMA 4.8. -

Suppose

^{that the}

function wH (h)

^{has a maximum at}

^hH.

Then

hH ho (The

^saddle

^So

^{is to the}

right of the

^maximum

of w~(h)).

Proof. -

^{The extrema of H- lie on}

~CH

^and

~C~

itself is a concave curve with a maximum at a

point

^{with an abscissa}

^hHH.

^Since

this

yields

that the maximum of ~C-

(if

^it

exists)

^{has an abscissa}

hH hHH

^{and moreover}

wH (h)

^for

h

Suppose

^that

^{ho hH.}

^{Then = and}

= > because >

wv(h)

^{for h >}

ho.

Therefore the intersection

point S*

^{of V and}

H h

^{has an abscissa}

^h*

^with

ho h* hH hHH -

^a contradiction with

(4.4).

This proves the lemma.

In order to make the facts

proved

in Lemmas 4.3-4.8 more

applicable,

below we list all needed

points

^{of H and V} with their coordinates and characteristics:

’

- the saddle

point, So

^{= H n}

V,

the

point

of maximum of

~+, Sv(hv , wv (hv)) -

^the

point

of maximum

of V,

the

point

^of maximum of

~-,

the

point

^of minimum of

~-C-,

~’(h, wH (h) ) -

^the

^point

^where

^H+

^{meets ~-~C’ .}

(The

last three

points

^{do not exist}

necessarily.)

Now we are able to describe in details the

picture

of the minima and maxima of

wo ( h ) .

LEMMA 4.9. -

(i)

^Let

Do

^>

0 for

any h. Then the

function

^{h 0}

has

exactly

^{one maximum and one minimum.}

(ii)

^Let

Do

⁰

for h

^E

(h-, h+),

^h-

^h+

^{0. Then}

a) for

h E

( - oc , h - ~

^the

function wo ( h )

^has

exactly

^one maximum and the

function wo {h)

^has

^exactly

^one maximum and one

minimum; b) for

h E

~h+, 0)

^the

function wo (h)

^has

exactly

^one maximum and one minimum.

Proof -

^The

separatrices

^{H and V} divide the half

plane h

0 into four open domains if

B2 -

3AE 0 and into five open domains otherwise. Let

us denote

by ~l~

^and

Vi

^the

_parts

^{of ?~C- and V}

respectively

^{which are to}

the left of the saddle

So

^and

by ^H2

^and

^{V2 -}

^{the rest}

part

^{of H and V.}

Analytically (for example)

^{we have}

= ~ ( h, wH ( h) ) :

^h

ho}

^and

similarly

for the other arcs. In the case where

B2 -

3AE ^>

0, ~-C2

^itself consists of two

separate

^{curves, one} of them is

H+

and the other is the

part

of to the

right

of the saddle. We will write sometimes and

~-C2

ⁱⁿ

order to

signify ^them,

^thus

H2

⁼

H+2

ⁿ

?nC2

^{in this case. Denote}

by Oij

^the

domain in h 0 which

boundary

^is

Hi

^U

Vj, i, j

⁼

1, 2. (More precisely

in the case

B2 -

3AE ^> 0

03A922

^{consists of two}

_separate

subdomains:

5222

with

boundary H-2

^U

^V2

^and

03A9+22

^with

^boundary U {h

⁼

0};

^{in the case}

B2 -

3AE 0 the

boundary

^of

^f222

^is

^~C2

^U

^V2 U {h

⁼

0~ .)

Therefore

depending

^{on the}

sign

^of

^{B2 -}

^the

position

of the saddle

point

^{and the}

availability

^{of extrema on ~-C ~ there are four cases I - IV}

(see Fig. 4, a,b,c,d):

I

B2 -

3AE

0,

^{~C- has no extrema,}

II

B2 -

3AE

0, So

is between

S~

^and

^S,

III

B2 -

3AE

0, So

is between

SH

^and

,~~,

IV

B2 -

3.4E > 0.

Annales de l’Institut Henri Poincaré - Analyse ^{non linéaire}